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19.—Improved Boundedness Conditions for Lowndes' Operators*

Published online by Cambridge University Press:  14 February 2012

Philip Heywood
Philip Heywood
Affiliation:
Department of Mathematics, University of Edinburgh
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Synopsis

The operator Ik(η, α) and its adjoint Kk(η, α) have obvious boundedness properties for α ≧½ because of their resemblance to fractional integrals. By expressing Ik(0, α) as the product of two Hankel transformations and a translation Heywood and Rooney [1] have shown that if 0<α<½ then Ik(η, α) and Kk(η, α) can be extended to bounded operators from a weighted Lp space to itself provided that 2/(1+α) ≦ p ≦ 2/(1−α) and the weight is suitably restricted. Heywood and Rooney conjectured that this p range could be improved, andin the present paper it is extended to

which may be best possible.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 73 , 1975 , pp. 291 - 299
Copyright
Copyright © Royal Society of Edinburgh 1975

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References

1Heywood, P. and Rooney, P. G., On the boundedness of Lowndes' operators. J.London Math. Soc, 10, 241–248, 1975.CrossRefGoogle Scholar
2Lowndes, J. S., A generalisation of the Erdé;lyi-Kober operators. Proc. Edinburgh Math. Soc, 17, 139–148, 1970.CrossRefGoogle Scholar
3Rooney, P. G., On the ranges of certain fractional integrals. Canad. J. Math., 24, 1198–1216, 1972.CrossRefGoogle Scholar
4Rooney, P. G., A technique for studying the boundedness and extendability of certain types of operators. Canad. J. Math., 25, 1090–1102, 1973.CrossRefGoogle Scholar